Course Outline

We are doing a reading course on optimal transport in the context of graphs and its ties with information geometry. A list of possible papers can be found here or for the summer, here. What follows are a list of relevant papers beyond these. When it is your week to present, please add any relevant papers to the list. To add your references, edit here.

Meeting time: Fridays at 1:30pm, 2240 MSB

April 12th, and 19th: Optimal Mass Transport: Signal processing and machine-learning applications

Presenter: Yiqun Shao

Relevant Papers:

38, Otto, 2001: the paper that presented the 2-Wasserstein space as a Riemannian metric derived from a Variational problem.

40, Park et al, 2018(Arxiv: The paper that transforms optimal transport into a Euclidian space in 1D via preserving distances to a reference distribution. It makes density functions separable after the transform.

Santambrogio 2015: Optimal transport for the applied mathematician. Good reference book for further reading.

50, W. Wang, D. Slepcev, S. Basu, J. A. Ozolek, and G. K. Rohde “A linear opti- mal transportation framework for quantifying and visualizing variations in sets of images,” The original LOT.

14, M. Cuturi “Sinkhorn distances: Lightspeed computation of optimal trans- port” Using the entropy penalty term to compute optimal transport map fast.

26, S. Kolouri and G. K. Rohde “Transport-based single frame super resolution of very low resolution face images”. Single frame superresolution using PCA on transport map.

April 26th: Quadratically-Regularized Optimal Transport on Graphs

Presenter: Shaofeng Deng

Relevant Papers:

D. Knowles Lagrangian Duality for Dummies!

Davis, T.A., 2011 This paper shows how to update the pseudo inverse of a graph Laplacian given a rank-1 change via Cholesky factorization.

Graph Laplacian A good tutorial on graph laplacian.

May 3rd: Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances

Presenter: Dong Min Roh

Relevant Papers:

T. Cover, J. Thomas; CH.2 Introduction to Entropy

P. Knight Sinkhorn-Knopp Algorithm

May 17th: Information geometry connecting Wasserstein distance and Kullback–Leibler divergence via the entropy-relaxed transportation problem

Presenter: Haolin Chen

Relevant Papers:

Amari 2016: Information geometry and its applications

Cuturi 2013: Sinkhorn distances: Lightspeed computation of optimal transport

May 24th: Information Geometry for Regularized Optimal Transport and Barycenters of Patterns

Presenter: David Weber

Relevant Papers:

Amari 2016 Recent reference on information geometry for machine learning by Amari.

Determinant of Block Matrices used in the proof of convexity of $D_\lambda$

Sherman-Morrison formula used in the proof of convexity of $D_\lambda$

Agueh & CarlieEnr, 2011 (preprint): Demonstration that the Barycenters of the Wasserstein metric are translation invariant.

Cuturi & Doucet, 2014: uses the C-function to compute Barycenters.

May 31st: The data-driven Schroedinger bridge

Presenter: Haotian Li

Relevant Papers:

June 19th: Application of the Wasserstein metric to seismic signals

Presenter: Bohan Zhou

Relevant Papers:

Engquist_2018_Seismic_Inversion: Data normalization techs for pre-processing seismic data.

Engquist_2018_Seismic_Imaging: A review for seismic inversion problem and the application of optimal transport.

Plessix_2006_Review: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications.

Yang_2017_Analysis: Analysis of misfit functions.

Engquist_2013_Application: Basic property to be satisfied before applying Wasserstein distance.

Bradley_2013_adjoint method: A manual for PDE-constrained optimization and the adjoint method from the beginning.

August 2nd and 9th: {Euclidean, metric, and Wasserstein} gradient flows:an overview

Presenter: Bohan Zhou

Relevant Papers:

Santambrogio_2017_Review: Gradient flow in Euclidean, metric, Waaserstein space giving three generic functionals including the heat equation, porous medium equation, Fokker-Planck equation and etc.

Ambrosio_2008_Gradient_Flow: The Bible in gradient flow.

Otto_2001_Geometry: Gradient flow strcture of porous medium equation. Compare the traditional approach to gradient flow of density with respect to L^2 norm of metric derivatives, with the new approach to gradient flow of density with repsect to Wasserstein distance, or equivalently the gradient flow of Lagrangian velocity with respect to L^2 norm.

Jordan_1998_JKO: JKO scheme is a discrete scheme, rising from Backwards Euler method, to show the weak solution exists systematically. Morevoer, the interpolated solution converges and it satisfies Fokker-Planck Equations in distributional sense.

August 16th and 23rd: Optimal Spectral transportation with application to music transcription

Presenter: Naoki Saito

Relevant Papers:

August 23rd and September 6th: A Parallel Method for Earth Mover’s Distance

Presenter: Haotian Li

Relevant Papers:

September 6th: Logarithmic divergences from optimal transport and Rényi geometry

Presenter: David Weber

Relevant Papers: